We prove that when a sequence of Lévy processes X(n) or a normed sequence of random walks S(n) converges a.s. on the Skorokhod space toward a Lévy process X, the sequence L(n) of local times at the supremum of X(n) converges uniformly on compact sets in probability toward the local time at the supremum of X. A consequence of this result is that the sequence of (quadrivariate) ladder processes (both ascending and descending) converges jointly in law toward the ladder processes of X. As an application, we show that in general, the sequence S(n) conditioned to stay positive converges weakly, jointly with its local time at the future minimum, toward the corresponding functional for the limiting process X. From this we deduce an invariance principle for the meander which extends known results for the case of attraction to a stable law.
"Invariance principles for local times at the maximum of random walks and Lévy processes." Ann. Probab. 38 (4) 1368 - 1389, July 2010. https://doi.org/10.1214/09-AOP512